Exercises: Similarity

A graphic designer is working on the logo for a new app. The original design is a rectangle measuring $8\text{ cm}$ wide and $12\text{ cm}$ high. For the loading screen, he needs to create a larger version that is similar to the original. If the width of the enlarged version must be $20\text{ cm}$, what will the height of the logo be on that screen?

On a house plan, the scale used is $1:50$. If the living room has a length of $12\text{ in}$ on the drawing, calculate how many feet the real length of this room measures.

Two triangles are placed in Thales's position, as shown in the following figure. Knowing that the horizontal segments are parallel, calculate the value of the unknown segment denoted by $x$.

A 10th grade student is $1,60\text{ m}$ tall. At a certain time of day, he casts a shadow on the ground of $2\text{ m}$. If right next to him, a street lamp casts a shadow of $5\text{ m}$, what is the height of the street lamp?

State whether the following pairs of triangles are similar and justify which criterion you have used (AA, SSS, or SAS):

• Triangle 1: Angles of $40^\circ$ and $70^\circ$.

• Triangle 2: Angles of $70^\circ$ and $70^\circ$.

A road map has a scale of $1:200.000$. If the real distance between two towns is $36\text{ mi}$, at what distance will they be on the map expressed in inches?

A pizzeria offers two sizes of round pizza that are similar. The "Small" pizza has a radius of $10\text{ cm}$ and an area of approximately $314\text{ cm}^2$. If the "Family" pizza has a radius of $20\text{ cm}$, what will its area be? (Remember the relationship between the scale factor $k$ and the areas).

A graphic designer is working on the logo for a new clothing brand. The original design is a quadrilateral with sides measuring $6\text{ cm}$, $8\text{ cm}$, $9\text{ cm}$ and $12\text{ cm}$. The client asks for a larger version for an advertising poster, where the side that previously measured $9\text{ cm}$ must now measure $45\text{ cm}$.

1. a) Calculate the scale factor $k$ between the poster logo and the original.
2. b) Determine the length of the other three sides in the poster logo.
3. c) If the perimeter of the original logo is $35\text{ cm}$, what will the perimeter of the poster logo be without adding its sides one by one?

On a house plan with a scale of $1:75$, the living room has a rectangular shape with dimensions of $8\text{ in}$ long and $6\text{ in}$ wide.

1. a) Calculate the real dimensions of the living room in feet.
2. b) Find the area of the living room on the plan (in $cm^2$) and the real area of the living room (in $m^2$).
3. c) Check that the ratio between the real area and the area on the plan is equal to the square of the scale factor ($k^2$).

To calculate the height of a communications tower that casts a very long shadow, Mark, who is $1,70\text{ m}$ tall, stands so that the end of his own shadow coincides with the end of the tower's shadow. If Mark is $25\text{ m}$ from the base of the tower and his shadow measures $2,5\text{ m}$, answer:

1. a) Why can we state that the triangles formed by Mark and the tower with their respective shadows are similar? Mention the similarity criterion used.
2. b) Calculate the height of the tower.

In the following figure, lines $r$, $s$ and $t$ are parallel. Use Thales's theorem to calculate the unknown lengths $x$ and $y$ expressed in inches.

A movie projector is located $6\text{ meters}$ from a screen. The light beam projects an image that has a height of $2,4\text{ meters}$ on the screen.

1. a) If we place an obstacle $40\text{ cm}$ high parallel to the screen to create a shadow puppet, at what distance from the projector must we place it so that its shadow occupies exactly the full height of the screen?
2. b) If we move the projector away to $9\text{ meters}$ from the screen, what will the new height of the projected image be?

Two rectangles are similar. The first has a base of $12\text{ cm}$ and an area of $96\text{ cm}^2$. The second rectangle has an area of $150\text{ cm}^2$.

1. a) Calculate the scale factor between the second and the first rectangle. (Hint: use the relationship between areas).
2. b) Calculate the dimensions (base and height) of the second rectangle.
3. c) What is the ratio of their perimeters?

Look at the right triangle in the figure. The altitude $h$ to the hypotenuse divides the original triangle into two smaller triangles.

1. a) Justify why the three triangles (the original and the two smaller ones) are similar to each other.
2. b) If the segments into which the hypotenuse is divided measure $m = 4\text{ cm}$ and $n = 9\text{ cm}$, calculate the altitude $h$ using the similarity of the two smaller triangles.
3. c) Calculate the area of the original triangle.

A furniture company manufactures regular hexagonal coffee tables. They have two models: the "Mini" model has a side of $30\text{ cm}$ and its glass top costs $45\text{ \$}$. The "Grand Lounge" model is similar to the previous one and has a side of $60\text{ cm}$.

1. a) What is the scale factor between the large and the small model?
2. b) Knowing that the price of the glass is proportional to its surface area, how much will the glass top of the "Grand Lounge" model cost?
3. c) If $180\text{ cm}$ of metal tape is needed to edge the small table, how much tape will be needed for the large table?

An architect is designing the facade of a modern auditorium. He has drawn a main right triangle and, to improve the acoustics, he needs to draw the altitude to the hypotenuse. This altitude divides the original triangle into two smaller triangles. Given that the legs of the original triangle measure $18\text{ m}$ and $24\text{ m}$, use the similarity of triangles to calculate:

1. a) The length of the altitude drawn to the hypotenuse.
2. b) The lengths of the two segments into which the hypotenuse is divided.

A 3D printing company has created a scale model of a circular plaza. In the model, the area of the plaza is $450\text{ cm}^2$. The client requests a second model where the perimeter is $20\%$ greater than that of the first model.

1. a) What will the area of the plaza be in this second model?
2. b) If the real plaza has a radius of $30\text{ m}$, what is the scale used in the first model? (Express the scale in the format $1:n$).

To measure the height of a cooling tower at a power station, a team of engineers uses a mirror placed on the ground. The engineer stands so that, looking into the mirror, he sees the reflection of the top of the tower. The mirror is $45\text{ m}$ from the base of the tower. The engineer, whose eyes are at a height of $1,75\text{ m}$, is $1,5\text{ m}$ from the mirror. Between the mirror and the tower, there is a security fence $5\text{ m}$ high.

What is the minimum distance the fence must be from the mirror so that it does not hide the reflection of the top of the tower from the engineer's eyes?

Two cities, A and B, are separated by a river with parallel banks. To calculate the distance between a point $P$ on the bank of city A and a point $Q$ on the bank of city B (located directly opposite), a surveyor walks $60\text{ m}$ along the bank of city A to a point $R$. From $R$, he walks another $20\text{ m}$ in the same direction to a point $S$. At $S$, he turns $90^\circ$ and walks away from the river until points $Q$ and $R$ are aligned with his current position ($T$). If the distance $ST$ is $15\text{ m}$:

1. a) Calculate the width of the river ($PQ$).
2. b) Calculate the straight-line distance between the surveyor at point $T$ and point $Q$ on the other side of the river.

A drone projects a conical beam of light onto flat ground. At a height of $10\text{ m}$, the beam illuminates a circle with a radius of $4\text{ m}$. The drone descends to a height where the illuminated area is reduced to exactly a quarter of the original.

Draw the diagram in your notebook.

1. a) At what height is the drone now?
2. b) If there is an object $1,5\text{ m}$ high in the center of the illuminated area, what will be the radius of the shadow cast by this object when the drone was at a height of $10\text{ m}$?

On a map of a forest area, a fire has affected a trapezoid-shaped area. The bases of the trapezoid on the map measure $4\text{ cm}$ and $7\text{ cm}$, and its height is $3\text{ cm}$. It is known that the actual affected area is $16,5\text{ hectares}$ ($1\text{ ha} = 10\,000\text{ m}^2$).

1. a) Calculate the scale of the map.
2. b) If it is decided to reforest an adjacent area which on the map is a triangle similar to the one that would be formed by extending the non-parallel sides of the trapezoid, and whose base is the shorter base of the trapezoid, what will be the actual area of this reforestation zone?

A slide is projected onto a screen located at a distance of $6\text{ m}$. The size of the projected image is $1,8\text{ m}$ high by $2,4\text{ m}$ wide.

1. a) If the original slide has dimensions of $24\text{ mm}$ by $32\text{ mm}$, at what distance from the slide is the projector's light source?
2. b) If we move the screen so that it is $9\text{ m}$ from the projector, how many times will the area of the image increase compared to the previous projection?

Consider an isosceles triangle $ABC$ where the unequal side $BC$ measures $12\text{ cm}$ and the equal sides measure $10\text{ cm}$ each. Inscribed in it is a square whose base rests on side $BC$ of the triangle and its other two vertices touch sides $AB$ and $AC$.

Use the similarity between the original triangle and the small triangle above the square to calculate the side of the square.